P/347 USA

Stability and Heating in the Pinch Effect

By M. N. Rosenbluth

One of the most promising types of thermonucleardevice is the stabilized pinch.1' 2 This consists of apinched cylindrical plasma in which a longitudinalmagnetic field is trapped. This field provides rigidityagainst the various types of instability to which thepinch is subject. An external conductor whichconfines the magnetic field of the pinch providesadditional stability, so that, with a proper choice ofparameters to define the equilibrium, the configurationmay be made linearly stable against all perturbations.

In the first part of this paper we shall discuss somesurface instabilities which may arise in the stabilizedpinch. In the second part we shall discuss the dis-assembly and heating of the plasma which resultsfrom collisions.

SURFACE-LAYER INSTABILITIES

Previous calculations of stabilitylf 2 have dealtmainly with an equilibrium with a very sharp surfacelayer such as might be expected with a very highlyconducting plasma.3 That is, for 0 < r < r0, we have

p = (1 + cxv2 - ap

2) (B0*/8n)ô, we have

Be = Boro/r; Bz = ayB0; p = 0.

(There is a glossary of symbols at the end of the paper. )The radius of the external conductor is ¡3r0. In the

thin region between r0 and r0 + ô, large surfacecurrents must flow. However, it would appear atfirst sight that in the limit of small ô this complicatedregion need not be considered explicitly, since, in thedynamic equations for the perturbation, one may usethe conditions of continuity of the normal compo-nents of the magnetic field and stress tensor to relatethe interior solutions to the exterior ones. We shallnow try to give a more complete discussion of theabove-defined problem.

It has been shown previously 4 that if the magnitudeof В does not change along a field line, as is the casehere, and if all distances are large compared with a

Be = 0; Bz =and for /3r0 > r > r0

* John Jay Hopkins Laboratory for Pure and AppliedScience, General Atomic Division of General DynamicsCorporation, San Diego, California. Research on controlledthermonuclear reactions is a joint program carried out byGeneral Atomic and the Texas Atomic Energy ResearchFoundation.

Larmor radius and we confine ourselves to the mar-ginal-stability case со = 0, then the standard magneto-hydrodynamic equations are valid. That is, thegoverning equations for the perturbation are

Vôp = [{VxB)xôB + (Vx<5B)xB]/4rc (1)

V-ЙВ = 0, (2)

where ÔB and dp are the perturbed quantities and Вis the equilibrium magnetic field.

The boundary conditions are regularity at theorigin and ôBr = 0 at r = ¡}r0. For the simplecylindrical geometry of the problem, the perturbationsmay be expanded in normal modes, i.e.,

dp, SB = (/>!, B1)ei(kZ+™e (3)

With this substitution, Eqs. (1) and (2) may beeasily reduced to a single equation as follows:

Let

then

where

and

X = rBlr/[kBz + [m\r)Be\

WFUir

¿ + k¿r¿ni

(4)

(5a)

г* (да2 + kh2)

d [ 2mBe[kBz

dr r(m2

{m/r)Be])(5b)

Since we are considering the marginal-stabilityproblem, it follows that if the solution of Eq. (4)which is regular at the origin should vanish at theexternal conductor, then the equilibrium is neutrallystable to the perturbation being considered. If thesolution crosses the axis before reaching the externalconductor, this implies instability, since we shouldhave to move the conductor in to stabilize the system.

85

86 SESSION A-5 P/347 M. N. ROSENBLUTH

For the general equilibrium specified by arbitraryBe(r) and Bz(r),Eq. (4) must, of course, be numericallyintegrated, although useful limits may be obtainedfrom the variational expression 5> 6 corresponding toEq. (4). A code for the numerical investigation ofEq. (4) has been prepared. However, for the sharp-layer case, i.e., ô <C r0, we may obtain analytic resultsusing the following procedure:

For r < r0 and r > r0 + ô, i.e., where there is nocurrent, Eq. (4) may be solved explicitly in termsof Bessel functions. This gives us values for thelogarithmic derivatives which must be joined on tothe solution in the surface layer.

Within the surface layer we can neglect 1/r comparedto д/dr and rewrite Eq. (4) in the form

where

and

и =

d~dr~

r2F

r0

2F (в)

dXB2X dr

{k2r2-m2)B2~2mkrBzBe

[m2 + k2r2)B2 [ }

In the region of interest, Be and Bz are changingrapidly. It is apparent from Eq. (5) that for someranges of k and m, depending on av, F(r) will passthrough zero, so that Eq. (6) develops a singularity.Parenthetically we may remark that because of thissingularity there is some difficulty about the deriva-tion of the magnetohydrodynamic Eq. (1). Thisquestion is receiving further study, but for the purposeof this paper we regard the conventional mode asvalid.

For modes which have no singularity, the right-hand side is negligible over the small distance ô, andwe find as the stability criterion

[и + Я] г, > [и (8)

For reference purposes, the explicit forms of thesefunctions are

Im{kr0)

[u + ЩГо+д = (m + «v

f* (h \ ^m\^o

P' (kX - — -

Km(kr0)o ; {kro)Im'(kro) (kro)Km'(kro)

where

Ge,m{kr0) =

1 ^>m(«^o)

Km'(pkr0) Im'(kr0)Km'{kr0) Im'(pkr0)

(9)

Equation (8) is, of course, identical with the oldstability calculation.

In discussing the singular case it is convenient toconsider the situation as we integrate towards the

singularity. Until we get very close to the sin-gularity—more specifically, until F ~ ô—the quan-tity и + H will remain constant, as before. Thisdetermines the value of и as we approach the singu-larity. Very near the singularity we may neglect theterm H ', so that in this region

и Jdr

= constant.

Hence, 1/u becomes infinite at the singularity as theintegral diverses. Moreover, depending on the sign ofи as it approaches the singularity, \\u may passthrough zero. This corresponds to X crossing theaxis, i.e., to instability. In order to avoid instabilityin this case we must then require that

[H]s -

r0 - [H]s

+ H],-a+s

0,

0.

(10a)

(106)

Here, [H]8 means the value of H at the singularity,which can easily be shown to be

[H]s =

It may seem strange that two criteria must besatisfied for the stability of a single mode. Theexplanation is that the singularity in Eq. (6) is sostrong that it completely separates the region interiorto the singularity from the exterior. Thus, if weviolate Eq. (106), for example, the instability isessentially confined to the outside of the surface layer.The new stability criteria are considerably morecomplicated than the old one, since they depend onthe structure of the surface layer. This structuremay be described by giving Be

2 as a function ofcp = tan-1 Be/Bz, the pitch of the field. Equation (10)then gives upper and lower limits for Be

2. It will benoted that the value of cp at the singularity is simplyrelated to k and m by kr/tn + tan <ps = 0. Therange of possible layer structures is further limitedby the requirement that the plasma pressure bepositive throughout. This gives

= B2 s i n 2 q>< BO

2{1 + ocY

2) s i n 2 < (12)

0.25.These results are plotted in Fig. 1 for ocv =As usual, we need consider only m = 0, ± 1 .

Curve I is the function и + H divided by a p

2

evaluated at r0. Curve I I is и + H evaluated atY0 + ô with various /} for m = ^ 1. The x's are them = 0 values of и + H evaluated at r0 + ô, evaluatedat k = 0, the most dangerous case. Equation (8),which must be satisfied for all k and m, hence all cp,and for both values of av, requires tha t Curve I I lieabove Curve I. This means tha t /9 — 2.5 is unstable.

The more stringent conditions (10) and (12) needonly be applied to values of k and m so that asingularity develops within the layer. This means,for the case of positive av, the region to the right of<p = t a n - 1 l/|(*v | and, for negative o¿y> to the left ofthis point. Condition (10) requires t h a t at

STABILITY AND HEATING 87

_

/'/'/

1i

ii /

/

/

/ /

Vy

У

ш //

N У

25

175

/

\ i

\

CM \

\

r

\

\\

\\\

347

Л

V\

Figure 1. Stability diagram for a v = ± 0.25 and/? = 1.1,1.75,2.5The shape of a thin surface layer is given by a graph of

B20/Bo

2 vs с?. To insure stability, this shape must be betweencertain limits, shown in this figure for the case a v = |25|,jft = 1.1, 1.75, 2.5. Previous 1>* stability criteria require onlythat curve I multiplied by a p

2 must lie above curve II for all qTbetween О and тт. In addition w e must now require for therange of "ф which occurs within the layer that the curve whichdefines the layer shape must itself lie between curve I (multipliedby a p

2 ) and curve II and above the cross. All physical layers(with positive plasma pressure) must lie below curve III. It canbe shown that curves III and II cross for all positive a v so that no

stable layer can be constructed

Be

2 must lie between Curves I and II and abovethe x. In addition, Eq. (12) requires Be

2 to bebelow Curve III. Hence, we deduce that there isno stable layer structure for ocv = + 0.25. Curve IVshows the type of layer which develops because ofdiffusion, as discussed below. This type of structurefor aY = — 0.25 is seen to be unstable against long-wavelength m = 0 modes, for the cases /5 = 2.5 and1.75, and against m = 1 modes with pitch close tothat of the external field.

It is, in fact, apparent that for positive OLY, i.e.,the external field in the same direction as the internalfield, no stable layers exist, since (10b) and (12) areincompatible. It appears that in most cases whereEq. (8) is satisfied and the field is reversed, somestable layers can exist. Moreover, their structure isnot too improbable although it appears that thediffusion layer is likely to be unstable against long-wavelength m = 0 disturbances and m = 1 dis-turbances near the pitch of the external field. Itwill be noted that condition (10b) appears much moredangerous than (10a), so that the instabilities arelimited to the outside regions of the plasma.

The importance of these instabilities is questionable.The bulk of the plasma is unaffected; moreover theyare slow—the multiplication time is proportional tothe surface layer width, and they cannot remain inthe linear phase of growth for long. It appearslikely that the surface will become unstable and thenrestabilize itself in a new equilibrium, perhaps helicalrather than cylindrical. It is noteworthy that suchhelical equilibria tend to generate a reversed Bz, asrequired for stability. Eventually, as we have noted,diffusion will make the layer unstable and the cyclemust be repeated.

While we have treated only the sharp-layer case,it is clear that the same qualitative features persistfor thick layers. The most important practicalquestion would seem to be under what conditions theinstability can proceed far enough to drive a signifi-cant amount of plasma into the walls, thereby releasingimpurities. Experimental work with a reversed Bzis in progress.

DISASSEMBLY AND HEATING

It is apparent that the stabilized pinch is not atrue equilibrium when interparticle collisions areconsidered. These collisions imply a finite conducti-vity of the plasma,7 so that electric fields must bepresent to drive the currents. These fields imply achange of flux, i.e., a relaxation of the crossed fields.At the same time, ohmic heating must occur, andtemperature gradients will lead to kinetic thermalconduction. As a consequence of these involvedprocesses, the plasma is simultaneously disassembledand heated. We shall find that a proper choice ofparameters leads to favorable conditions for a thermo-nuclear reaction with no external heating mechanism.Qualitatively, this is because the crossed-field con-figuration represents a higher energy state of themagnetic field than the state which will prevail afterdiffusion has occurred and the components of the fieldshave been mixed. The excess energy goes intoheating the plasma. Of course, a quantitativeanalysis is required to estimate the magnitude of theeffect and to compare the disassembly and heatingrates.

We therefore study the evolution of the equilibriumsituation described at the beginning of this paper. Adetailed solution will be presented for the case withthe initial conditions ô = 0, a sharp layer; ap & 1, alow-temperature plasma; and av = 0, a 90° anglebetween the fields. It is apparent from the diffusioncharacter of the problem that early departure fromthe initial values is confined to the neighborhood ofthe surface. For analytic simplicity we shall thereforeconsider a plane problem, calling the initial surface-layer position X = 0 and letting the plasma extendextend — oo and the vacuum field to + oo.

The equations of motion 8 are : Maxwell's equations ;the conservation of mass, momentum, and energy forthe plasma; Ohm's Law; and the equation governingheat transport in the plasma.

These are

8B

V x B =

2V{QkT) = j x B

(13)

(14)

(15)

(16)

88 SESSION A-5 P/347 M. N. ROSENBLUTH

— {3QkT + V . (0 + 5çkTv) - E - j = 0 (17) ю8t

ib*!*2Б2(7

Q =me

(В X VT) (18)

(19)

a =(kT)ï

ж e2(mGï)lnA= G (kT)\ (20)

Here, E is the electric field, B the magnetic field,j the current density, g the number density of ions orelectrons, k the Boltzmann constant, v the plasmavelocity, m\ and me the ion and electron masses, andЛ the ratio of maximum to minimum impact para-meter.

In Eq. (16) we have used the fact that the difíusionvelocities are very subsonic, so that hydrostaticequilibrium is maintained. (Needless to say, we arenot considering possible unstable motions.) It isassumed throughout that the distribution functionsare close to Maxwellian, with equal temperatures forelectrons and ions. This is easily shown to be thecase after the difíusion wave has penetrated a distancelarge compared to an ion Larmor radius. Equations(18) through (20) are, of course, valid only within thislimit. They have been derived 8 from the transportequation by considering the small deviations from theMaxwell distribution which are necessary to com-pensate for the collision terms in the presence offield, density and temperature gradients. A con-venient mathematical tool for this purpose is theFokker-Planck Equation.9

For a plane problem with an initially sharp layer,the system of equation has a similarity solution of thevariable Xj^lt. The resulting ordinary differentialequations have been integrated on the GeneralAtomic IBM-650 by Miss G. Roy. The solution isshown in terms of the appropriate dimensionlessvariables in Fig. 2.

The abscissa is the dimensionless similarity variable

I = itF

Here, X is the Lagrangian coordinate representing theinitial position of the mass point, Bo and Q0 are thefield strength and particle density in the undisturbedregions, (p is the angle the field makes with the originaldirection of the field in the plasma, and /? is thedimensionless pressure, /3 = \6лдкТ/В0

2.It will be seen that the plasma pressure reaches a

maximum value of about 0.43 near f = 1. The field isabout half uncrossed at this point. For f < 1 thesolution is nearly isothermal; for f > 1 the densityremains almost constant. In the following discussionwe will refer to f = 1 as the depth of penetration of

\ р/У

д/ \

L. "

о 16 тгр к ТjK= i

^ ^

____________

3)" ^УГ '

" • - —

_ •

04

02

Figure 2. Solution of diffusion Equations 13-20 for the casedp = 1 , 0¿v = 0

The results of diffusion resulting from a case in which theplasma pressure is initially low compared with the magnetic pressureand in which the magnetic field in the vacuum is at right anglesto the field in the plasma. Plasma pressure, plasma density,and magnetic field direction are plotted vs X/V¿ where X is the

Lagrangian distance from the original interface

the wave, and denote quantities at this point by thesubscript p.

The following features of the solution seem of somesignificance.1. The stabilized pinch needs only to be preheated toa temperature consistent with the recombinationrate (about 100 ev). From this point on, the heatingoccurs naturally through intermixing of the fields.The plasma comes to a temperature

(21)

As mentioned earlier, this heating does not occur foruncrossed fields.8

2. The plasma-vacuum interface moves a distance ofabout 1.2 Xv into the space initially occupied by thevacuum field. Hence, in a typical pinch configurationno plasma will touch the wall until the wave haspenetrated quite deeply.3. The front of the wave is characterized by a nearlyconstant shear. As discussed in the first part ofthis paper, the resulting layer shape is linearly un-stable against only a rather narrow region of wave-length perturbations if a negative Bz is provided.4. The rate of penetration depends on the conductivityat the temperature of the heated plasma. In cgsunits, with the temperature in kilovolts,

(22)

5. Similarly, the energy delivered to the plasma is

(23)

The other important mechanisms in the energybalance of the burning pinch are fusion and radiative

STABILITY AND HEATING 89

loss; i.e., bremsstrahlung.energy according to

dE

Both of these produce

(24)

Thus, if the temperature is greater than a criticalvalue, about 6 kev for D-T and 50 kev for D-D, thereaction is self-sustaining.10 It may be well tooperate only slightly above this critical value to thatdisassembly is not substantially faster than thatgiven by Eq. (24). At a fixed temperature, the dis-assembly time is proportional to r0

2, as may be seenfrom Eq. (22). Moreover, using Eqs. (24) and (21)we see that the burning time is proportional to B0~

2.Hence, we should expect the efficiency to be a functionof Boro only, i.e., the current. In fact, it can beeasily shown that if the quantities B0

2/Q0 and Boroare held fixed, the complete set of equations fordisassembly, heating, and burning can be simplyscaled to a change in r0. We may note that thecurrents must be at least 2 X 106 amp to contain thea particle resulting from the D-T reaction.

Finally, we indicate in Table 1 a possible D-Treactor design, taking an optimistic criterion fordisassembly, i.e., Xv = r0. The figures should betaken as only a rough indication.

Table 1. Possible Characteristics of a Diffusion-limited,Self-heated D-T Reactor

Quantity Value Scaling

Major radius of torus 30 cm (arbitrary) r0

Minor radius of torus 6 cm r0

I n i t i a l p l a s m a r a d i u s ( r 0 ) . . . 1 . 5 c m r 0

Current 3 X I06 amp (ro)°Pressure at wall 400 at m ro~

2

Init ial pinched densi ty (Q0) . . 1.3 x 1017/cm3 ro~2

B u r n i n g t e m p e r a t u r e (Tp) . . . 6 k e v (ro)°

D i s a s s e m b l y t i m e 0.15 sec r02

T o t a l m a g n e t i c e n e r g y . . . . 4 x 1 0 6 j o u l e r0

Losses (copper torus) 2.5 x 106 joule r0

Energy produced 3 x 107 joule r0

T e m p e r a t u r e r ise of c o p p e r sur-face d u e t o r a d i a t i o n . . . . 500°C ro~

2

Efficiency of burning

ACKNOWLEDGEMENTS

Dr. James Alexander has done the numericalwork in connection with the theory of surface-layer-instabilities. The author wishes to thank him andDr. Norman Rostoker, both of General Atomic, formany useful discussions.

The qualitative features of the intermixing heatingwere realized independently by S. A. Colgate andJ. L. Tuck. It is a particular pleasure to acknowledgesome stimulating discussions with Dr. Colgate andwith A. N. Kaufman, who independently had studiedsome of these questions.8

Bz

PÔ

k

m

X,F,u,HЦХ),

9a

f

Tv

Qo

K(X)

GLOSSARY OF SYMBOLS

radius of pinchunperturbed azimuthal fieldunperturbed longitudinal fieldvalue of azimuthal field, Be, at r0

ratio of internal longitudinal field, Bz,toB0.

ratio of external longitudinal field, Bz,to Bo. o¿y is defined as positive if theexternal field has the same sign as theinternal one

(as used in first part) ratio of externalconductor radius to r0

plasma pressurethickness of surface layerlongitudinal wave number of per-

turbationazimuthal wave number of perturbationperturbed magnetic fielddefined in Eq. (4)defined in Eq. (7)Bessel functions in Watson's notationvalue of Be

2 at the radius where F(r) = 0

plasma conductivity(as used in second part) 16nQkT/B0

2

a dimensionless variable defining thedepth of penetration

temperature at f = 1undisturbed plasma density

Mr. Rosenbluth presented a survey, at the Conference,of Papers P/347 (above), P/354, P/1861, P/376 andP12433:

Linear stability theory in a magnetohydrodynamic,collision-dominated fluid is a fairly well understoodsubject.11 However, m a high-temperature plasmain which the particles interact only through themacroscopic fields, the situation is not so clear. Letus begin by discussing the types of waves characte-ristic of an infinite homogeneous plasma with constantmagnetic field.

The situation is shown in Table 2. The magneticfield is taken to be in the Z direction, and we considera wave propagating in the X, Z plane. The dis-tribution function is an arbitrary function of themagnitude of velocity and its component parallel tothe field. In general, four types of wave appearpossible. One of these is a trivial mass flow in thedirection of the field which causes no charge or

90 SESSION A-5 P/347 M. N. ROSENBLUTH

Table 2. Types of Plasma Waves

В = Bz; E = E exp [i (cot -f ¿XX + kzZ)]; f = / (г;2, v z

Character Stability

Plasma oscillation . .Electrostaticсо «¿ сор = л/(47гпе21те)

Landau damped

Overstability occurs if groups ofelectrons have a vz substantiallyexceeding mean thermal velocity.

Transverse

Alfvén waves . . . . E» Overstable for long wave-lengthsif ion distribution is anistropic.

Undamped- Incompressible

Hydromagnetic waves

Р , 2 Б 2

Unstable if — - > --P у OTT

or P« - P , >?- 2

current. The other three are indicated in Fig. 3with a qualitative discussion of their properties. Thecomments are not inclusive or exact.

If the distribution function is isotropic and adecreasing function of energy, it is easy to show ongeneral statistical grounds that the plasma must bestable.12 However, it appears that even in theinfinite homogeneous case, small deviations in thedistribution function may lead to alarming instabilities.

For the plasma oscillations, overstability, i.e.,growing oscillations, occurs for a wavelength suchthat kzU = cop, where U is the velocity of a non-thermal group of electrons, or of electrons and ionsrelative to each other.13 Thus, these particles movein phase with the disturbance. It will be noted thatthe resonant particles see a non-oscillatory electricfield and hence can move across magnetic field lines.

In the event that plasma heating is produced byelectric fields parallel to the magnetic-field lines, wemay expect that groups of high-energy electronswill be readily created because of the fall-off of crosssection with energy. Thus, such a parallel fieldleads naturally to an unstable situation. The Alfvén-wave instability depends on a similar resonance forсо + kzvz = Larmor frequency.13 It is particularlynoteworthy since it occurs even for very small pressureanisotropies.

The hydromagnetic waves, which are the mostnearly analogous to ordinary sound waves, becomeviolently unstable for large pressure anisotropies.14

In particular, a shock perpendicular to the field linescreates a large transverse pressure, P±. The resulting

UNSTABLE STABLE

Figure 3

instability may play an essential role in producingthe entropy necessary for the existence of the shock.Thus we see that there are many unstable situationseven for a simple infinite plasma.

When we consider a finite geometry, the situationis, of course, much more complicated. Progress todate has been made largely by assuming that thecharacteristic scale of inhomogeneities is large com-pared to Larmor radius and Debye length and thatfrequencies are small compared to orbital and plasmafrequencies. As we have heard in the paper presentedby M. D. Kruskal,15 a variational expression forstability with these approximations may be obtainedwhich does not differ substantially from the fluidcase. It should be noted that the plasma oscillationand Alfvén instabilities which we have discussed donot appear in this approximation. Thus, there hasbeen no complete theoretical demonstration that anyfinite confined plasma can be stabilized.

However, even in this magnetohydrodynamicapproximation it is difficult, although possible, toattain a stable static equilibrium. Perhaps thesimplest situation to be studied is one in which theplasma contains no internal magnetic field. Thisconstant-pressure plasma is then confined by anexternal magnetic field which must obey the con-dition that B2/8TZ = pressure along the surface. Ithas been shown that a necessary and sufficient con-dition for stability is that the principal normal to thesurface must at all points be directed into the plasma.16

This result is not restricted to the linear theory.The geometry is illustrated in Fig. 3. The shadedregion in the diagram represents the plasma. Onthe left we have a convex unstable surface. This isthe pinch. The cusp device on the right is concaveat all parts and therefore stable. It appears to bethe only possible confined stable equilibrium in whichthere is no field embedded in the plasma. Un-fortunately, there is a finite rate of plasma loss throughthe cusps.

Another situation which has received a preliminarystudy is that of a plasma supported against gravityby a rotating magnetic field. The rotation slowsdown the instability but does not eliminate it.16

STABILITY AND HEATING 91

/л ^TRANSITION SHEATH

PLASMA

Figure 4. Stabilized pinch geometry

It is perhaps fair to say that the geometry whichhas received the most attention throughout the worldis the stabilized pinch. It was realized early thatthe pinch was subject to the above-mentionedinstability.17 In order to correct the situation, anaxial magnetic field was introduced into the plasma.This situation is shown in Fig. 4. It was then shownby several authors that complete stability could beobtained by proper choice of the internal field andthe position of an external conducting shell.18 How-ever, these treatments neglected the structure of thecurrent-carrying surface layer in which the fieldchanges its direction. In practice, this sheath maybe thick.

The equations of motion governing a perturbationof the form ег(кг+тв) may be easily written. Theseequations develop a strong singularity at the radiuswhere kBz + (m/r)Be = 0, that is to say, at such aradius that the pitch of the perturbation matches thespiral of the unperturbed magnetic field. This isperhaps not surprising since at this radius the per-turbation will not bend the field lines appreciablyand the plasma can flow freely along the lines.

The mathematical effect of the singularity is thatthe region exterior to the singular point is completelyseparated from the interior of the plasma. Thisbrings into existence a class of surface instabilitieswhich are not affected by the internal stabilizingfield. The essential character of the instability is anazimuthal bunching of the parallel current filamentswhich exist at a given radius.

A useful necessary condition for stability 19 is thatat all points

Bz2 r

~8лГ 4 [*- 4dr

Necessary and sufficient conditions have been foundfor the favorable case of a very thin transition layer.20

It can be shown that only very special shapes of surfacelayer are stable. In particular, no stable surface canbe found unless there is a longitudinal field outsidethe plasma which is opposite in direction to theinternal field.

Let us now consider the diffusion and intermixingof the crossed magnetic fields due to interparticlecollisions—without regard to stability. From theRutherford cross section one can compute the varioustransport coefficients of the plasma—electrical andthermal conductivities and thermoelectric coefficients.Then the usual conservation equations—mass, mo-mentum and energy—plus Maxwell's equations provide

a complete set of dynamical equations for the diffusionprocess.20 In particular, we study the case of aninitially sharp surface layer which separates a regioncontaining low-pressure plasma and axial magneticfield from a vacuum region with azimuthal magneticfield; i.e., the stabilized pinch. For the early periodof the diffusion, a plane approximation is adequate.In this case, the equations may be solved in terms ofa similarity variable proportional to original distancefrom the interface divided by the square root of time.

The results are shown in Fig. 5. The abscissa is thedimensionless similarity variable; £ = 0 is the initialinterface. The scale is such that | = 1 is about theskin depth which one would estimate using theconductivity deduced from the temperature whichexists at | = 1. Q/QQ is the plasma density relativeto its initial value ; ф is the pitch angle of the magneticfield ; and /? is the ratio of material pressure to magneticpressure. The significant feature of the results isthat most of the energy liberated by uncrossing thefields is delivered to the plasma,21 raising its pressureto about 0.43 of the initial magnetic pressure, regard-less of the initial pressure. Hence, there exists avery efficient mechanism for creating very hightemperature plasmas.

Figure 5. Solution of the dynamical equations for the plasmadiffusion and field intermixing

Finally, we may inquire as to the eventual resultof the diffusion process. If no external electric fieldis applied, the plasma will, of course, diffuse outwardindefinitely until lost to the walls. On the otherhand, an applied axial electric field is capable ofcausing a drift to balance out the collisional diffusion ;and, in fact, detailed solutions have been found fora steady state of this type in collisional equilibriumaway from the walls.22 Unfortunately, this equili-brium is very unstable hydrodynamically.

To sum up, recent theoretical work has shownthat the stabilized pinch is a self-heating device. Onthe other hand, the existence of surface instabilitieswill require a careful programming of magneticfields. In addition, the large electric fields used inpinch formation may well lead to the formation ofunstable plasma waves.

92 SESSION A-5 P/347 M. N. ROSENBLUTH

REFERENCES

1. M. N. Rosenbluth, The Stabilized Pinch, p. 903, Proc.of the Third International Conference on IonizedGases, Venice (1957) ; a more complete version isavailable as Report LA-2030 (1956). Declassified.

2. R. J. Tayler, The Influence of an Axial Magnetic Fieldon the Stability of a Constricted Gas Discharge, p. 1067,Proc. of the Third International Conference on IonizedGases, Venice, (1957) ; Tayler has also studied theuniform current case. There has also been considerableSoviet work by Shafranov and others (cf. Réf. 18).

3. M. N. Rosenbluth, A. W. Rosenbluth and R. L. Garwin,Infinite Conductivity Theory of the Pinch, Report LA-1850(1954). Declassified.

4. M. N. Rosenbluth and N. Rostoker, Theoretical Structureof Plasma Equations, P/349,this Volume, these Proceedings.

5. B. R. Suydam, private communication.6. E. A. Frieman, private communication.7. L. Spitzer, Physics of Fully Ionized Gases, Chap. 5,

Interscience Publishers, New York (1956).

8. M. N. Rosenbluth and A. N. Kaufman, Plasma Diffusionin a Magnetic Field, Phys. Rev., 109, 1 (1957).

9. M. N. Rosenbluth, W. M. MacDonald and D. L. Judd,Fokker-Planck Equation for an Inverse Square Force,Phys. Rev., 707, 1 (1957).

10. R. F. Post, Controlled Fusion Research, Revs. ModernPhys., 28, 338 (1956).

11. I. B. Bernstein, E. A. Freeman, M. D. Kruskal andR. M. Kulsrud, Proc. Roy. Soc, A244, 17 (1958).

12. W. Newcomb, to be published.13. K. Wilson and M. Rosenbluth, to be published; I.

Bernstein, to be published.14. M. Rosenbluth, LA-2030 (1956), declassified; S. Chan-

drasekhar, A. Kaufman and К. М. Watson, Ann. Phys. 2,435 (1956).

15. M. D. Kruskal and C. R. Oberman, On the Stability ofPlasma in Static Equilibrium, P/365, this Volume, theseProceedings; M. Rosenbluth and N. Rostoker, TheoreticalStructure of Plasma Equations, P/349, this Volume, theseProceedings.

16. S. Berkowitz, H. Grad and H. Rubin, Problems inMagnetohydrodynamic Stability, P/376, this Volume, theseProceedings.

17. M. D. Kruskal and M. Schwarzschild, Proc. Roy. Soc.18. M. Rosenbluth, LA-2030 (1956); R. J. Tayler, Proc.

Roy. Soc, B70, 1049 (1957); V. D. Shafranov, J. NuclearEnergy, 2, 86 (1957).

19. B. R. Suydam, Stability of a Linear Pinch, P/354, thisVolume, these Proceedings.

20. M. Rosenbluth, Theory of Pinch Effect—-Stability andHeating, P/347, this Volume, these Proceedings.

21. This mechanism was suggested by S. A. Colgate.22. C. L. Longmire, The Static Pinch, P/1861, this Volume,

these Proceedings.